Algorithms versus Circuit Lower Bounds

Different techniques have been used to prove several transference theorems of the form "nontrivial algorithms for a circuit class C yield circuit lower bounds against C". In this survey we revisit many of these results. We discuss how circuit lower bounds can be obtained from derandomization, compression, learning, and satisfiability algorithms. We also cover the connection between circuit lower bounds and useful properties, a notion that turns out to be fundamental in the context of these transference theorems. Along the way, we obtain a few new results, simplify several proofs, and show connections involving different frameworks. We hope that our presentation will serve as a self-contained introduction for those interested in pursuing research in this area.